chatper 3 sol
TRANSCRIPT
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New Century Maths 11
Mathematics General
Preliminary course (Pathway 2)
Worked Solutions
Chapter 3
SkillCheck
1 a
25 36
61
7.810...
7.81
+
=
=
b
81 9
72
8.485...
8.49
=
=
c
2.551... 2.55
d
151.260... 151.26
2 a
2500 10 25 000 mm =
b
1400 1000 1.4 m =
c
2.5 1000 2500 mm =
d
9 100 900 cm =
e
960 000 1000 960 m =
f
3600 100 36 m =
3 a
180
b
360
4 a0.4245
b0.7547
c 0.9563
d 1
e0.0175
f0.5
g0.5
h0.0998
i0.8111
5 a
18 18 6 3
30 30 6 5
= =
b
15 15 3 5 21
9 9 3 3 3
= = =
c
( )12 : 28 4
3: 7
both by
=
d
( )27 :18
3: 2
both by 9
=
e
( )3 5
:4 8
6 : 5
first one by 2
=
f
1
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( )
15 cm: 40 mm
150 mm : 40 mm
15:4
both by= 10
=
g
( )
1 :10
100 :10
10:1
m cm
cm cm both by= 10
=
h
( )
50 km : 500 m
50 000 m : 500 m
700:1
both by= 500
=
i
1 cm : 2 m
1 cm: 200 cm
1:200
=
=
6 a
24
2 5
242
5
48
5
39
5
h
h
h
h
=
=
=
=
b
183
18 3
54
k
k
k
=
=
=
c
284
4 28
28
4
7
p
p
p
p
=
=
=
=
7 a
72 90x + =(complementary angles add to 90)
18x =
b
35y =
(alternate angles in parallel lines)
c 200 360k+ = (angle o re!ol"tion is 360)
160k =
d
130 180a + =(s"pplementary angles add to 180)
50a =
Exercise 3!1
1 a
2 2 22 5
29
29
5.385...
5.39 m
b
b
= +
=
=
=
b
2 2 2
2 2 2
8.1 3.5
8.1 3.5
53.36
53.36
7.304...7.30 m
v
v
v
= +
=
=
=
=
2
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c
2 2 210 7
149
149
12.206...
12.21cm
q
q
= +
=
=
=
d
2 2 2
2 2 2
24 15
24 15
351
351
18.734...18.73cm
p
p
v
= +
=
=
=
=
e
2 2 24 4
32
32
5.656...
5.66 cm
h
h
= +
=
=
=
f
2 2 2
2 2 2
9.4 5.5
9.4 5.5
58.11
58.11
7.622...7.62 m
k
k
k
= +
=
=
=
=
2 a
2 2 2
2 2 2
10 4
10 4
84
84
9.165...
9.2cm
z
z
z
= +
=
=
=
=
b
2 2 2
2 2 2
61 60
61 60
121
121
11.0 cm
c
c
z
= +
=
=
=
=
c
2 2 218 24
900
90030.0
t
tcm
= +
=
=
=
d
3
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2 2 285 50
9725
9725
98.615...
98.6 m
w
w
= +
=
=
=
e
2 2 216 63
4225
4225
65cm
x
x
= +
=
=
=
2 2 2
2 2 2
87 65
87 65
3344
3344
57.827...
57.8 cm
y
y
z
= +
=
=
=
=
f
2 2 2
2 2 2
15 10
15 10
125
125
11.180...
x
x
z
= +
=
=
=
=
2 2 2
2 2 2
24 10
24 10
476
476
21.817...
y
y
z
= +
=
=
=
=
11.180... 21.817...
32.997...
33.0 m
a x y = +
= +
=
3
2 2 23.8 1.8
17.68
17.68
4.204...
4.20 m
d
d
= +
=
=
=
4
2 2 2
2 2 2
482 140
482 140
212 724
212 724
461.22...
461m
d
d
d
= +
=
=
=
=
5
2 2 2
2 2 2
6 2
6 2
32
32
5.656...
5.66 m
h
h
d
= +
=
=
=
=
4
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6
2 2 2
2 2 2
200 160
200 160
14
14
120 na"tical miles
x
x
x
= +
=
= 400
= 400
=
7
2 2 2
2
270 1050
1175 400
1175 400
1084.158...
1080 m
x
x
x
= +
=
=
=
" a
2 2 2
2 2 2
2
3.5 1
3.5 1
11.25
11.253.354...
3.35 m
h
h
h
h
= +
=
=
==
b
2 2 25
11.25 25
36.25
36.25
6.020...
6.02 m
l h
l
= +
= +
=
=
=
#
2 2 2
2 2 2
25 17
25 17
336
336
18.330...
18.3m
18.3 1.1
19.4 m
x
x
x
h
= +
=
=
=
=
= +
=
1!2 2 220 7.5
456.25
456.25
21.36...21.4 cm
l
l
= +
=
=
=
11
2 2 20.7 3.9
15.7
15.7
3.962...
3.96 m
p
p
= +
=
=
=
12
5
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2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
7 19
410
410
19 17
650
650
22 4
500
500
7 10
149
149
29 10
941
941
#erimeter
410 650 500 149 941
110.986...
110.99 m
AB
AB
BC
BC
CD
CD
DE
DE
EA
EA
= +
=
=
= +
=
=
= +
=
=
= +
=
=
= +
=
=
= + + + +
=
13 a
2 2 2
2 2 2
$ase 20
15 12
15 12
81
81
9
$ase 9 9 20
38cm
perimeter 38 15 20 15
88cm
0.88m
0.9m
x x
x
x
x
= + +
= +
=
=
=
=
= + +
=
= + + +
=
=
b
2 2 2
2 2 2
"nkno%n side
70 50
70 50
2400
2400
perimeter 50 2 2400 2
197.979...
198.0 m
y
y
y
y
=
= +
=
=
=
= +
=
14
2 2 2
2 2 2
2.4 2
2.4 2
1.76
1.76
1.326...
1.3m
x
x
x
= +
=
=
=
=
15
2 2 22.5 0.45
6.4525
6.4525
2.540...
2.54 cm
x
x
= +
=
=
=
Exercise 3!2
1&atc'ing sides # and *
6
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8scale actor 2
4
+
= =
2
20 2
30 3
BG mm
P mm
= =
3
4
3
!"
EA=
4,eac'er to c'eck
5 * and + angles not necessarily e-"al as only
kno% t%o o t'em.
* and e-"iang"lar ratio matc'ing sides /
2
3
6 a and
3 5 6 1
6 10 12 2= = =
b* and
9 5
12.6 7=
c and
6.55 2.5 1
13 5 2= =
7
48scale actor 9.6
52 m 9.6
19.2 m
h
= =
=
=
"
2.90.358...
8.1
5.2 0.358...
1.861...
1.86m
1.9m
scale factor
h
= =
=
=
=
#
265.2
5
3 5.2
15.6m
scale factor
h
= =
=
=
1! a,'e "n is lo%er.
b
15 m 15 mscale actor 0.0025
6 km 6000 m
1500 0.0025
3.75m
h
= = =
=
=
11
4.8scale actor 3.2
1.5
lengt' s'ado% 1.8 m 3.2
5.76 m
= =
=
=
12
10.5 m 10.5 mscale actor 43.75
24 cm 0.24 m
30 cm 43.75
1312.5 cm
13.125m
h
= = =
=
=
=
13
2.3scale actor 0.123...
18.6
9.4 m 0.123...
1.162...
1.16 m
hei#ht
= =
=
=
14
7
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150scale actor 6.25
24
%idt' o canal 15 m 6.25
93.75 m
= =
=
=
15
3.2 3.2scale actor 0.372...
3.2 5.4 8.6
lengt' o s"pport 12.5m 0.372...
4.651...
4.7 m
= = =+
=
=
16
32 m 32 mscale actor 45.714...70 cm 0.70 m
'eig't 30cm 45.714...
13.714...
13.7 m
= = =
=
=
Exercise 3!3
1 a
1cm : 1km
1cm : 1000 m
1cm : 100 000 cm1: 100 000
=
==
b
mm : 1cm
2 mm : 10 mm
2:10
1:5
2
=
=
=
c
cm : 1000 m
4 cm : 100000 cm
1:25000
4
=
=
d
10 cm : 1m
10 cm : 100 cm
10:100
/ 1:10
=
=
e
m : 2 cm
500 cm : 2cm
500:2
250:1
5
=
=
=
2
6 cm 6 000
000 000 cm
/ 360 000 m
/ 360 km
000
= 36
o .
3
55 km 2 000
km
/ 0.0275 m
/ 2.75 cm
000
= 0.0000275
4 a
cm : 120 km
3 cm : 120 000 m
3 cm : 12 000 000 cm
1: 4 000 000
3
=
=
=
b i
8
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2.1cm 4 000
00000cm
/ 84 000 m
/ 84 km
000
= 8 4
ii
6.1cm 4 000
00000cm
/ 244000 m
/ 244 km
000
= 24 4
iii
cm 4 000
00000cm
/ 216 000 m
/ 216 km
5.4 000
= 21 6
i$
cm 4 000
00000cm
/ 120000 m
/ 120 km
3 000
=12 0
c
92 km 4 000
km
/ 0.023m
/ 2.3 cm
&ittagong
000
= 0.000023
d
km 4 000
km
/ 0.01m
/1 cm
atoom$a and #enrit'
or &ittagong and ollongong
40 000
= 0.00001
5 a
4 100
400 cm
/ 4 m
=
b
2 4 0.5 cm =or 5 mm
c
9 3000
27000 cm
/ m
=
270
6 a
9.2 cm $y5.5 cm
b
cale 1cm:10 m
1cm:1000cm
1:1000
=
=
=
o cm 1000
5500 cm
55 m
5.5
=
=
c
o cm 1000
9200 cm
92 m
9.2
=
=
7
7.4 2000.037 m
3.7 cm
*
=
=
" a
5 4 20 km =
b
3.3 4 13.2 km =
c
2.9 4 11.6 km =
9
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# a
1cm:3m
10cm:30m
lengt' 30 m
=
b
cm:3m
0.9 cm : 2.7 m
%idt' 2.7 m
1
=
c
1cm:3m
7.5cm:22.5m
lengt' 22.5m
=
1! a
1cm 200
200 cm
2 m
=
=
b
2
1.4 cm 200
280 cm 2.8 m
cm 200
480 cm 4.8 m
*rea 2.8 4.813.44 m
= =
2.4
= =
=
=
11 a
2.2 cm $y 2.5 cm
4.4 m $y 5 m
=
b
24.4 5 6 3 40 m + =
c
1cm 200
200 cm
2 m
=
=
d
2
cm 200
440 cm 4.4 m
cm 200
200 cm 2 m
*rea 4.4 2 8.8 m
2.2
= =
1
= =
= =
e
2
cm 200
300 cm 3 m
cm 200
360 cm 3.6 m
*rea 3 3.6
10.8m
+ost 10.8 34
367.20
1.5
= =
1.8
= =
=
=
=
=
Exercise 3!4
1
1cm:2.5m
/1cm:250cm
/1: 250
scale actor 250.
=
a
0.8 cm 250
200 cm
2 m
=
=
b
2.4 cm 250
600 cm
6 m
=
=
10
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c
2
ot' lengt' and %idt' meas"re 2.4 cm
*rea lengt' %idt'
6 m 6 m
36 m
=
=
=
d
12
2
indo% dimensions:
0.6 cm 250 150 cm 1.5 m
0.4 cm 250 100 cm 1 m
*rea / ront $ack 2 side %indo%
/ 6 2.5 6 2 2 ( (2 2.5) 6) 1.5 1
15 12 27 1.5
52.5 m"m$er o litres o paint
52.5/
10
5.25
need 6
= =
= =
+ + +
= + +
=
=
.
2cale actor / 400
a1.5 cm 400
600 cm
/ 6 m
=
b
2 cm 400
800 cm
/ 8 m
=
c
2
%idt' / 1.5 cm 400
600 cm
/ 6 m
*rea / 8 6
48 m
=
=
d
cost / 48 84.50
4056
=
3 a 4
b%alkin ro$e
c1
d kitc'en
e
2
2.9 3 4.83 3.99
8.7 19.2717
27.9717
28 m
+
= +
=
4 a 7
c
2
8.4 1.8
15.12 m
=
d
2
*rea 1 tile 0.3 0.3
0.09 m
15.12 0.09 168 tiles
=
=
=
e
2
4.2 3.3 6.5 4.0
13.86 2639.86 m
cost 39.86 86
2710.48
+
= +
=
=
=
5 a1:100
b
11
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13.5 cm 100
1350 cm
13.5 m
=
=
c12 %indo%s
d 2 doors
e
2
2
6.4 1.2
7.68 m
8 m
=
f
2400 mm 2.4 m=
g
2
2.2 1.25 5.2 0.8
2.75 4.16
6.91 m (ans%ers may !ary
slig'tly depending on meas"rements)
cost 105 6.91
725.55
700
+
= +
=
=
=
6 a
3600 mm $y 3600 mm
/ 3.6 m 3.6 m
b i3.6 m
ii
2
*rea 3.6 3.6 3.6 0.6
12.96 2.16
10.8 m
=
=
=
c
2
2
lengt' 2.7 2.7 1.7 3.7 3.6
14.4 m
%idt' 1.2 m
*rea / 14.4 1.2
17.28 m
17 m
= + + + +
=
=
=
d
2
15 (7.8 1.2)
15 9
135 m
+
=
=
e
cost / 135 1540
207 900
=
f i
2.7 0.2 13.5 tiles
14 tiles
=
ii
14 14 196 tiles =
Exercise 3!5
12
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4 a
sin 8 1924.90...
25
b
cshift a=
=
o
b
n 3.2692
72.99...
73
A shift ta=
=
o
c
cos 3 6
60
bcshift a =
= o
d
n 0.4129
22.43...
22
$ shift ta=
=
o
e
sin 0.9815
78.96...
79
% shift=
=
o
5 a
cos 0.3368
70.3179... ;